Any plane not passing through O , inverts to a sphere touching at O.
A circle, that is, the intersection of a sphere with a secant plane, inverts into a circle, except that if the circle passes through O it inverts into a line. This reduces to the 2D case when the secant plane passes through O , but is a true 3D phenomenon if the secant plane does not pass through O. The simplest surface besides a plane is the sphere. The first picture shows a non trivial inversion the center of the sphere is not the center of inversion of a sphere together with two orthogonal intersecting pencils of circles.
The inversion of a cylinder, cone, or torus results in a Dupin cyclyde. A spheroid is a surface of revolution and contains a pencil of circles which is mapped onto a pencil of circles see picture. The inverse image of a spheroid is a surface of degree 4. A hyperboloid of one sheet, which is a surface of revolution contains a pencil of circles which is mapped onto a pencil of circles. A hyperboloid of one sheet contains additional two pencils of lines, which are mapped onto pencils of circles.
The picture shows one such line blue and its inversion. This mapping can be performed by an inversion of the sphere onto its tangent plane. They are the projection lines of the stereographic projection. The 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the Cartesian coordinates.
One of the first to consider foundations of inversive geometry was Mario Pieri in and The point at infinity is added to all the lines. The invariant is:. According to Coxeter,  the transformation by inversion in circle was invented by L. Magnus in Since then this mapping has become an avenue to higher mathematics. The combination of two inversions in concentric circles results in a similarity , homothetic transformation , or dilation characterized by the ratio of the circle radii.
The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space. However, inversive geometry is the larger study since it includes the raw inversion in a circle not yet made, with conjugation, into reciprocation.
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Inversive geometry also includes the conjugation mapping. As mentioned above, zero, the origin, requires special consideration in the circle inversion mapping.
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In the complex number approach, where reciprocation is the apparent operation, this procedure leads to the complex projective line , often called the Riemann sphere. It was subspaces and subgroups of this space and group of mappings that were applied to produce early models of hyperbolic geometry by Beltrami , Cayley , and Klein. Thus inversive geometry includes the ideas originated by Lobachevsky and Bolyai in their plane geometry.
inversive geometry - Wiktionary
Furthermore, Felix Klein was so overcome by this facility of mappings to identify geometrical phenomena that he delivered a manifesto, the Erlangen program , in Since then many mathematicians reserve the term geometry for a space together with a group of mappings of that space. Narasimham Narasimham Thanks for the answer. In a partial answer to the one above it I can only say Basically inversions are a homothety or scaling in a similitude situation, I shall reply in EDIT1 soon, Or do these generalised circles straight lines meet at infinity ,similar to the original circles that meet at the center of inversion?
It is the same question as above but I included a figure to make the question more clear? Does the larger circles lines also touch each other at one point like the smaller circles before inversion?
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Email Required, but never shown. Featured on Meta. Fukagawa, H. Gardner, M. Jeans, J. The Mathematical Theory of Electricity and Magnetism, 5th ed. Cambridge, England: The University Press, Johnson, R. Boston, MA: Houghton Mifflin, pp. Kelvin, W. Principles of Mechanics and Dynamics, Vol. New York: Dover, p. Lachlan, R.
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Schmidt, H. Die Inversion und ihre Anwendungen. Munich, Germany: Oldenbourg, Thomson, W.